Sun, 20 Feb 2000 11:25:14 +0200
Hasan Keler <nx at cheerful.com>
wrote:
> I asked:
>> > "ZFC is a foundation of the 99% of mathematics."
> >But what is the remaining 1% ?
> >And why is ZFC not a foundation of this 1% ?
.........
> Also Matt Insall writes:
>> >Mathematicians ... seem to act as though everything they do
> >could, and someday will, be encoded in ZFC.
>> Naturally, the following question comes to me:
>> Is there a theorem of "ordinary" mathematics that can NOT be
translated into
> the language of set theory ?
First note that theorem assumes a theory in which it is proved.
Therefore the question should be not about a single theorem,
but rather on a theory.
Really, ZFC at present plays the role of a canonical form of the
contemporary mathematical reasoning and of a mathematical language.
However, as I already mentioned in FOM, there is in principle a
possibility of some unusual, "non-ordinary" mathematics which is
not translatable into (the language of) set theory. I believe that
mathematics in general is a science on various formalizations
(of anything what deserves and capable to be formalized).
Therefore nobody could guarantee that any possible formalism
(and corresponding intuition behind of it) may be translated into
(reduced to that of) set theory. One such example is set theory NF
which is unknown to be reducible to ZFC. Another example
(evidently not reducible to ZFC) was given by R.Parikh
[Feasibility in Arithmetic, JSL, 1971]:
Parikh demonstrated by the ordinary (metamathematical) methods
that his theory PA_F extending Peano arithmetic by the predicate
F for an "explicitly" bounded infinity of *feasible numbers* is
"almost" consistent: he estimated a shortest proof of contradiction
in PA_F by so big number that this surely guarantees that nobody
would be able to write *explicitly* (symbol-by-symbol) a
contradiction.
(What is the reason to consider and formalize new mathematical
concept of feasible number - is a separate question.)
According to the principle of Hilbert discussed recently on FOM
(Consistent => Existing; I would say Consistent => Imaginable;
I essentially agree with the comments of Robert Black
<Robert.Black at nottingham.ac.uk> Date: Fri, 18 Feb 2000 00:10:05 +0000
on this subject, in particular, I agree that this leads to a kind of
mathematical Platonism - rather harmless one - (local or relative)
Platonism of a working mathematician - the term of Karlis Podnieks;
by the way, the comments of Black strongly demonstrate that he
himself
is not a (global) Platonist and I would advise to anybody here to
reread
his posting)
(almost) consistency of PA_F means that intuitively there "exists"a
world of natural numbers with an *infinite* but "explicitly" *bounded*
initial segment F without the last number. (This is stronger than the
usual feature of non-standard models of arithmetic.) At least, we could
imagine this world. And this is a possibility for a new mathematics
with new kind of intuition which is not reducible to set theory.
(PA_F has no model in ZFC universe! G"odel's completeness theorem
does not work here.)
Thus, there are essentially two sources for this 1%:
(1) Extending ZFC by new stronger and stronger axioms, i.e.
working in the framework of and in the language of ZFC, and
(2) considering arbitrary (probably almost consistent) formalisms
different from (versions) of ZFC.
I believe that (2) may contain a lot of new interesting mathematical
ideas. Therefore I prefer the definition
(100% of) Math = FS (Formal Systems)
instead of more restrictive
(99% of) Math = ZFC.
The latter definition does not explain what are the missing 1%.
This definition rather neglects these 1% or may be 0.00001%.
"Math. = FS" gives more possibilities for extending these 1%
outside ZFC and this extension could be in principle very
interesting and reach. We should not close this possibility by
the definition of Math. in terms of ZFC. Also "Math. =FS",
if explained in more details, gives an alternative view on the
nature of mathematics and mathematical formalisms strongly
different from Platonism. We need only the normal "non-global"
*Platonism of working mathematicians* (when working in
classical first order theories). However note, that new formalisms
not reducible to ZFC may be even very unlike to the ordinary
first-order theories what means that even "local" or "relative"
Platonism in general has not so many chances to survive; only
general intuition of unpredictable form is necessary and possible.
Vladimir Sazonov
Appendix:
PA_F is
PA (Peano arithmetic in the language containing symbols
of sufficiently many primitive recursive functions) +
the following axioms for new predicate F for "feasible" numbers
(not allowed to participate in the Induction Schema)
0,1 in F;
F+F contained in F (therefore F is intuitively "infinite");
F is down closed (x<y & y in F => x in F);
t is not in F (i.e. F is upper bounded by t) for some closed
fixed prim. rec. term t denoting a very large number.
This number should be really very huge. Otherwise
PA_F would be contradictory by a short proof. However,
it proves to be possible and rather interesting to replace this
formalism in favor of some more "realistic" upper bound t
for feasible numbers. In difference with PA_F, the latter
formalism is rather unlike to the ordinary first-order theory.